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Notoriously, quantum computation shatters complexity theory, but is innocuous to computability theory.Yet several works have shown how quantum theory as it stands could breach the physical Church-Turing thesis.
Gandy formulates postulates about physics, such as homogeneity of space and time, bounded density and velocity of information --- and proves that the physical Church-Turing thesis is a consequence of these postulates. Thus this approach exhibits a formal non-trivial interplay between theoretical physics symmetries and computability assumptions.
To me this seems a little wishy-washy; a Turing machine can approximate the problem to an arbitrary degree of precisions, just as a pen-and-paper approach is only capable to executing to a finite degree of approximation.
Still, an "idealized" ruler and compass does provide a framework for describing an algorithm which is not directly Turing computable.
Slightly more in detail, the (physical) says vaguely that what is computable in the mathematical sense of computation is precisely what is “effectively” computable (physically computable).
In interpreting this one has to be careful which concept of computation is used, there are two different main types: Indeed, there are physical processes (described by the wave equation) which are not type-I computable (Pour-El et al.